Reading a global P travel-time curve

PHYSICS · Subjective · Fermi University Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Rays and travel times: from arrivals to a layered planet

Problem

Reading a global P travel-time curve, someone measures the local slope dT/dΔ=11.38 s deg−1\mathrm{d}T/\mathrm{d}\Delta = 11.38\,\mathrm{s\,deg^{-1}} at Δ=18∘\Delta = 18^\circ, and takes from a traced ray that the corresponding turning radius is rt=5956 kmr_t = 5956\,\mathrm{km} (depth 415 km415\,\mathrm{km}). Someone argues: "at the turning point sin⁡i=1\sin i = 1, so the spherical ray parameter is p=rt/vtp = r_t/v_t. Using p=11.38p = 11.38 and rt=5956 kmr_t = 5956\,\mathrm{km}, vt=rt/p=5956/11.38=523 km s−1v_t = r_t/p = 5956/11.38 = 523\,\mathrm{km\,s^{-1}}." Find the error in this reasoning and give the correct turning speed vtv_t.

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